THE ELEMENTAI,Y THEORY OF ABELIAN GROUPS Introduction

57 Pages·2002·3.05 MB·English

ANNALS OF MATltFMATICA_ LOGIC o. Volume 4, No. 2 (1972) 115-171
THE ELEMENTAI,Y THEORY OF ABELIAN GROUPS
Pzml C. EKLOF *
Stanford l'nirersiO', Stanford, Ca/if 94305, U.S..4.
and
Edward R. FISCHER
"haY Unircrsity, Aew ltaren, Corn'., U,S,,I.
Received 24 June t971
Introduction
Ill this paper we make a compretiensive survey of the first-order pro-
perties of abelian groups. The principal method is the investigation of
saturated abelian groups. As a result of our determination of the struc-
ture of saturated groups we are able to give new model-theoretic proofs
of the results of W, Szmietew 12 ; moreover we obtain new results on
the existence of saturated models of complete theories of abelian groups;
and we also generalize our results to modules over Dedekind domains.
One of tile principal results of Szmielew is the determination of group-
theoretic invariants which characterize abelian groups up to elementary
equivalence (The decidability of the theory of abelian groups follows re-
latively easily from this result), Now elementarily equivalent saturated
groups of the same cardinality are isomorphic; so our method is to look
r~_if invariants which characterize saturated abelian groups up to isomor-
phism. We prove that any ~:-saturated group A(tc _> ~ ) is built up in a
specified way from the groups ZQ~'), Zp, Z(p n) and ~ and that the
number of copies of these groups which occur are determined by the
elementarily definable dimensions dim(p n- ! Ap ), dim(p n- l A/onA ),
and dim(p n=l Ap/pnA p ) and by the exponent of A (for explanations
of the notation and more details, see § l ). These dimensions, which arise
Re.arch p~tia!!y sponsored by the Air Force Office of Scientific Research, Office of Aero-
ecap~. Rescav.-h, United S~tes Air Force. undo," AFOSR Contract No. F44620-70-C-OO7Z
116 P.C. Eklof and E.R. Fisher. The elenventaO, theoo" f~¢ nait,.~ba groups
naturally in considering the structure of saturated groups, are, respective-
ly, the Szmielew invariants 3~p ) P, n ~ (A), ~Op IP, n I (A), and o, )3~ lp. n I (,4).
Thus we are led to ~zmlelew s theorem. In fact, we work with slightly
different invariants viz.
D(p,A) = lim dim(/,"A lpl ),
which determines the number of copies of the divisible group Z(p ~ ) in
a saturated group A;
,)Al+"p/A"p(mid
Tf(p;A) = lim
which determines the number of copies of the torsion-free group ~zZ in
A;
Anp/lpi
and U(p, n -~- 1 ; A ) = dim(p ~n ! A lt'l ),
the Ulm invariants, wtlich determine tile number of copies of ZIp ~' ) in A.
We begin in section 0 by reviewing tile theory of v.aturated structures
and take advantage of tile opportunity to distinguish carefully between
two notions of homogeneity which have sometimes been conlMsed in
the literature. Section 1 deals with the structure of ~:-saturated groups:
in section 2 we prove the criterion tbr elementary e,itfivalence of groups
and tile decidability result and also give a criteriot~ for elementary em-
bedding of groups which is implicit in i 2. In section 2 we also prove
that any group which has the structure described in section 1 is in fact
K-saturated. Using this we are able, in section 3. to ~ietermine precisely
the cardinals in which a given complete theory of abelian groups has sat-
urated models. The table in section 3 summarizes this information and,
gives a complete analysis of the categoricity and ~l-stability of complete
theories of abelian groups. In section 4 we give a model-theoretic proof
of the existence of an elimination of quantifiers for a conservative ex-
tension of the theory of abelian groups. Finally in section 5 we general-
ize many of the preceding results to theories of modules over Dedekind
domains.
We would like to thank S, Feferman r~Li some helpful conversations
on the subject of elimination of quantifiers. We are also gra~eful to Eli
Bers for helping to clear up our confusion about the notion of ho~,loge-
neity.
§ ,O Preliminaries 1 l 7
§ .O Preliminaries
In this section we will review some of the basic lacts about saturated
and hot~logeneous-universal models, For even mort: basic model-theoret-
ic notions such as "'elementary substructure", "formula", "l-type", etc.,
we refer the reader to Shoenfield l I 1 , Bell and Slomson 1 , and Chang
and Keisler 3. All results not otherwise cledited can be found in the
papers of Morley and Vaught 91 and Vaught ! 31.
We will use latin capitals A, B, C, .... to denote sets and German capi-
t~tls, ,~ ~, ff ..... to denote structures on these respective sets. A struc-
ture 9 is ,,;aid to be for a language L or to be an L-structure if {~ has an
interpretation for each non-logical constant of L (and no other relations,
functions, and individualsJ, In later sections, when we deal with abelial~
groups we will follow custom and use latin capitals for both sets and
structures. We think of a language as the set of (first-order) formulas in
its vocabulary: thus, a language always has infinite cardinality, x and
will be used exclusively to denote cardinals, and Card(A) will denote the
cardinality ofA. 9~(A) stands for {S c A ICard(S) < x}.
Given a structure I~ we say that 19 is elementarily x-universal or (since
no confusion can arise for our purposes) just x-universal if every struc-
ture ~ which is elementarily equivalent to .lP and of cardinality less than
:~ can be elementarily embedded in ,{~ lP is said to be (elementarily) -c1
homogeneous if every isomorphism between elementary substructures
of 19 of cardinality less than :~ can be extended to an automorphism of
Given subsets S and T of str,~ctures lP and ~ respectively for a com-
mon language k we say that a function f: S ~- T is a local elementary
isomorphism if for every formula ¢(v~ ..... v n ) of the language L and
every s I ..... s,) ~ S.
iP ~¢ls t ..... 1%. iff ~3~¢f(s l) ..... f(s n) .
Clearly such a local elementary isomorphism admits a unique extension
to a local elementary isomorphism between the substructures generated
by S and 7 i.e., the closu~s of S and T under the fup~tions (if any)
named i0 ~L
A structure lP is said to be poitltwise g-homogeneous if for every pair
118 .C.P Ektof and E.R. Fisher, e~.T elementao.' t,~x~" of abelian $v~p~
S, T~ 9~(A), element a ~ A, and local elementary i~xmlorphism f:S ~ T,
there exist a' ~ A and a local elementary isomorphism g :S u {a} -~-
T o {a'} which extendsl:
A ~-expansion of a language L is a language which is obtained from L
by the addition of < :~ new individual constants, A ~:-expansion of an L-
structure is defined analogously. We .~ly that a structure t~ is ~.saturated
if for every ~-expansion '~19 of 9!, .q' reMizes ',2'reve type of Th(Pl' .~ An
elementary theory T is said to be ~c-stable if for every ~:-expansion 'I)~ of
a model fi~ of T there are < :~ l-types in Th(?').
A structure ~ is said to be universal if it is Card(A )*-oniversal, homo-
getwous if it is Card(A )-homogeneous, pohttwise homogeneous if it is
pointwise Card(A )-homogeneous, and saturated if it is Card(A )-saturated.
These notions are related through the following well-known theorems.
We assume throughout that .~!~ '~, and ~ are L-structures and that
:~ _> ~ = Card(L) -< Card(A ).
.O 1. Theorem. ?f is r~-saturatcd iff 1? 's po#ttwise ~-homogeneous and
~ *-universal.
0.2. Theorem. fI? is saturated i tf ~.~ is pohttwise homogeneous and uni-
versal.
0.3. Corollary. Any two el~ ,nentarily equivalent saturated structures of
the same cardinaliO' are isomoq~hic
0.,1. Theorem. If ? is po#ltwise ~-hom~eeneous attd a E A. then eveo"
local elementary isomorphism between subsets S and T of A s,tch that
Card(S) = Card(T) < ~ cJn be extended to an isomorphis:n beiween ~ie ~
mentary substrttctures '~ and ~ of fI? such tha~ a E B.
0.5. Theorem. lfC,~rd(A ) > ~, then ? is saturated (ff l,~ "~,i hom~eeneous
and universal.
It should be clear that every structure ?l tbr a countable language
such that Card(A ) = ~-t 0 is homogeneous, but l~ot every .~ach structure
is pointwise t~omogeneous. We now give a counter-example, due tO
Morley, to the possibility that these two notions coincide !br um-ount-
able structures in countable languages.
§ ,O seira,imtler~l I i 9
Let A = 0, I u {- 1}. For each positive intege~ n, let U n be the sub-
set of A consisting of those real numbers in 0, I which have a 1 in the
n-th place under some,binary expansion (the binary rationals -¢ 0 have
two binary expansions), Let ~ be an equivalence relation on A such that
0~-IandforallaEA,a~0ora~.-l,and{ala-0} and{ala--l}
are both dense in 10, I , We claim that Z19 = (A, -, U l , U z .... ) is homo-
geneous, but not pointwise 2-homogeneous.
Indeed, let f: 1~, ~ ~ be an isomorphism between elementary sub-
structures of .IP Then f must clearly ieave every element of B n (0, 1
fixed. Also, if 0 E B, then there exists a ~ B n (0, 1 such that 0 =- a.
Then .f(O) =- f~a) = a, and f{O) = .O Similarly if --- 1 ~ B, then f(- 1 ) = - .1
In any case, '8 = ~ and f is the restriction of the identity automorphism
of ,1~ On the other hand, 0 and - 1 satisfy the same l-type. This is per-
haps best seen by taking a saturated elementary extm:~ion *t~, of i~ and
constructing an automorphism of *I)* which takes 0 to -I.
0.6. Theorem, Let iP be ~'-tmiversaL and h't od be a set of fbrmulas such
that .fi~r all ¢ E od the ,l?ee rariabh,s of os art, in ~v{ ~I < ~.}. Then if od is
consistent with Th(PD, od is realized by a ~¢-sequence in .9
Proof. Let Z be the result of replacing the free variables in ~ by new in-
dividual constants {ct : ~ < .}¢~ Then Th(PD u Z has a model of cardina-
iity .¢I By ~¢*-universality, this model can be elementarily embedded in
,lP and the theorem is proved.
The previous theorem is particularly useful in conjun, tion with the
next one.
0.7. Theorem. tf 19 is ~-saturated. then erery ~-expansion of ~ is ~-sat-
uratcd.
0.8 Theorem. If P is ~¢-saturatcd. then ere,3' re&wt of ~? is ~-saturated,
0.9, Theorem. tf 9 is ~¢-saturated ar, d ~ "si a substructure of IP such that
B = {a ~ ,4 1 ~t =1 ~ola} forsome,fornmla ~, then ~2 is to-saturated.
0. I 0. Theorem, Erery structure has a tc-sagurated elementary extension.
021 .C.P E,~lof and E.R. Fisher. The ¢lementao, theoo, of abelian s~qiorg
Corollary 0.3 above shows that if there exists a ~turated model of a
complete theory T with cardinality :~ ~ Card(T), then it is unique up to
isomorphism, Existence is provided by the following theorem,
Let x~ = E~ < ~ ~:~, and call a cardinal K a saturation cardinal if ~ = ~.
0.11. Theorem. Every theoo, T having hafinite models has a saturated
model in every saturation cardinal ~ > Card(T).
Every strongly inaccessible cardinal and every cardinal such that
= *~ = 2 x is a saturation cardinal; moreover, it is consistent to assume
that 2 ¢0, for example, is very large and yet is a ~turation cardinal.
The reader who needs a justification for an appeal to the existence of
arbitrarily large saturated models lbr a theory T can follow one of four
routes. First, he can assume that there are arbitrarily ~arge inaccessibles.
Seco~ldlyo he can assume something like the generalized continuum hy-
pothesis, Thirdly, he can assume no more thaq the usual axioms fi~r set
theo:.7 and employ an argument involving the construct ible universe.
For example, suppose that l?' c ~, and that one can show that tP <- 3/' if
a saturated elementary extension of ~ exists in some cardinal :~ > Card(Bj.
Let T' be Th(2~') where '~.~ is an expansion of ~ in which every element
of B has a name. Let S be the ~ransitive closure of {'~l', T'} and let L(S)
be the G6del universe obtained by constructing sets in the usual way but
starting with S instead of the ~:~pty set. in L(S) it is true that
Kf~tV -> Card(S) ~- 2 ~ = ,)*~t
Reasoning in L(S) we conclude painlessly that ~9 <- ~ holds in L(S) and
therefore absolutely, i,e., in V. A last approach which tile reader can at-
tempt to follow is to modify the original proof and use ~-saturated
structures which are not saturated: in other words, a rich supply of local
elementary automorphisms must replace the automorphism in the origi-
nal argument.
0.12. Theorem. Let ,~ and ~ be L-structures, amt let ~ be a saturation
cardinal -> Card(A ) + C~lrd(B) + Card(L)'. Then the~turated elementary
extensions '? of l~ a~ld '~' of '~ of cardinality :~ are such that '~ ~ ~'
is a saturated elementary extension of ~ X '~.
~, O, t~l~minories 121
Proof. We may assume that A, ,11 and A x B are pairwise disjoint. Let
5t "-- (A X B 3~ A u B, VA× R, V A , \"B" rq , rr2, ,,.),
where VA×/~, V A , and ~/V are unary relations holding for the mem:~ers
ofA × B, 4, and B respectively, rr l and rr 2 are the projections ,'q : ,t × B-*
,4, rr 2 : A × B ~ B, and the remaining relations are those of 9 x ,5~' ,19
and ~. Now a saturated elementary exte~sion if' of ~ exists haw~g
cardinality :~ and the obvious relaIivized reducts are readily seen to be
elementary extensions of 19 X ~,~, ,fO and ~ which are saturated by
Theorems 0.8 and 0.9 above.
,O 13. Corollary l Feferman-Vaught 41 ). Giren L-str;wtures ,1~,~ IP 2, ~1 a.,td
~k 2, (t'91 ! -= ~? and ~1 -~ '~2 thcn Ptl X 1~\ ~ 2tP X ~2. and if
The method used in the proof of Theorem 0.12 can be employed to
show that the theory of the limit of any finite diagram (in the category-
theoretic sense) is determined by the theories of the structures in the
diagram.
Our last application will be needed in a later section.
0.14. Theorem (Macintyre 7 ). If ~ and ~ are L-structures such that
Th(?l) and Th('.~)are r,-stabtefor K > Card(L)~, then Th(9I X Bk )is -:~
stabh,.
Proof. We may assunlc that Card(A ) = Card(B) = a saturatien cardinal
¢t and float 9 and '~ are saturated. Since 9 × ~ is saturated (by 0.1 2)
it suffices to show that, for every subset SofA × B with Card(S) < ,ot
there are fewer than ¢n 1-types realized in the ~¢-expansion of 9 × ~s in
which the members of S have names. We may assume that S is of the
form S t X S 2 with ~S c A and S 2 c B. Define (a, b) ~ (a', b') ira satis-
fies the same type as a' in the S l -expansion of t~, and b sati.~fies the same
type as b' in the S2-expansion of ~. The number of equivalence classes
is clearly le~ than ,¢I while on the other hand, given (a, b).~ (a', b'),
there exist automorphismsf ! and f2 of ~1~, and B,' which fix the members
orS 1 and S 2 and move a to a' and b to b', respectively. 1"3 × f2 thus fixes
221 P.C. Eklof and E.R. Fisher, The elemenlao, theo O' of abetian groups
the members ofS = S l X S 2 and moves (a, b) to (a', b'), showing that
(a, b) and (a', b') realize the same type in the S-expansion of 19 x ~. and
completing the proof.
The following two results will be useful m a later section.
0.15. Theorem (Vaught ! 13 ), An co I -stable theory in a countable lan-
guage has a countable saturated model,
Iffl " 1~ "~ ~1 and f2 : 9,l, ~ ~_~ are elementary monomorphisms,
we say thatf l is equivalent tof 2 if there exist isomorphismsg: tl~,
~2 and h: ~l ~ ~-~¢ such that
.6
I~i i l . .... > '.8)
g h
J 2"
commutes.
0.16. Theorem. Let T be a theoo' in a countable language. Suppose that
there exists a countable set S of triph, s (P, f. '~ ) such that every elemen-
tary monomorphism g: ?l' "- ~' between countable models of T is eqtd-
valent to some f: ~,~ ~- '~ wit;: (~A,', ~ ) E S. Then T is O~l-Stable.
Proof. If T were not co I -stable there ~ould exist a countable model iP of
T with an oc I -expansion ~' such that uncountably many I-types m'e in
Th(,~I'). Putting together the facts that (i) every 1-type can be real;zed
in some countable elementary extension of ,~' (ii) only countably many
types can be realized in a given countable elementary extension, and
(iii) equivalent elementary extensions realize the same types, we get the
desired contradiction.
§ .1 The smwture osaturated ataqian spuotvt, 123
§ 1. The structure of saturated abelian spuorg
Our goal is to characterize all the co~hplete theories of abelian groups.
This problem is equivalent (giver the existence of saturated models of
every complete theory) to characterizing the isomorphisna classes of sat-
urated groups (of ~ fixed cardinalityL In order to solve this problem,
we proceed, through a series of ~emmas, to give a complete structure
theory, aml to isolate a complete set of elementarily definable invariants,
for saturated abelian groups.
Most of the algebraic tools we need can be found in the very readable
book by Kaplansky 16. We will review many of the alg~ braic definitions
and theorems we use, but we refer the reader to 6 ~ for ,urther details.
~Rel\'renccs in parentheses are to 6 unless otherwise specified). We be-
gin by recalling some fund:mlental definitions. First. "group" will always
mean abclian group, if A is a group and n -~ Z, nA = {na:a ~ A}. Ifp is
a prime, A Ipl = {a .~ A :pa = 0}. (We will write nA p instead of (nA)p).
lfa E A, u ~ Z, we ~y n divMcsa (we write nta) if there exists b E A
(not neces.,~rily unique) such that bt~ = .a A is called di~,isible if for every
aE A and every n :~ 0, n divides a (§5). A subgroup B of A is called a
divisible subgroup ofA ifB is a divisible group in its own right. IfB is a
divisible sub~oup of A, B is a direct summand of A (Theorem 2). A is
called reduced if A has no non-zero divisible subgroups.
Tluxmghout this section we wilt be working towards the goal of deter-
mining the structure of a ~-saturated group S, where ¢t is an uncountable
cardinal, However, some of the results along the way will be true under
weaker hypotheses, which we will point out for the purposes of later
sections, In fact our first structural results use only the fact that S is to l-
equationally compact i.e, any cotmtable system of equations (in any
number of unknowns) with constants from S which is finitely solvable
m S is solvable in S, tt is known that a group is co ! -equationally compact
if and only if it is a direct summand of every group in which it is con-
tained as a pure subgroup ( 5 , Theorem 38.1 and Exercise 5, p. 162).
Such groups are called pure-injective (or algebraically compact) and their
structure is known (51, Prop, 40,1). However in order to keep our dis-
cussion as sell-contained as possible we will give a simple direct analysis
of the structure of an w l-equationally compact group S.
421 P.C, Eklof and E.R. Fisher, The elementao" theory f~- na~l¢ba spm~rg
Let S a be the maximal divisible subgroup of S i.e. the union of all tile
divisible subgroups of S. We may write S = R @ S a where R is reduced
(Theorem 3). S a is uniquely determined; R is not uniquely deter:nined
in general but it is unique up to isomorpifism because it is isomorphic to
S/Sa.
Let S r = S/Sa: we will frequently.witl~out further ten, ark ide~til:y S r
with a (not uniquely determined) subgrt~up of S so that $ = 3; ~ S d. On
occasion it will be neces~wy to identify "5 r with a more cxplici~!y defined
subgroup of S.
Let D = the set of all elements orS ~hich are divisibk by every inte-
ger 4: 0. Clearly S a ~ D. but the opposite inclusion is false in general.
However it is true in an w I -equationally compact group:
1.1. Lemma. Let S bean co I -equationally compact gro~q~, Then S d. the
maximal divisible subgrotq~ of S, equals D. the set of all ele,~,nts orS
divisible by every htteger 4: .O
Proof. D is clearly a subgroup of S. To prov, D 2g S,t it suffices to prove
D is a divisible subgroup ors Given a ~ D, n ~ Z ~ {0}, consider the
set of equations 6 = {tn v m = x :m ~ Z - (0}} u {nx = a}. C is finitely
solvable in S; indeed, it suffices to prove that ibr any m ~ Z -- {0},
my m = x and nx = a are simultaneously solv-'.b~e in S, which is clear since
mn divides a ~ D. Therefore C is solvable in S i.e. there exists b ~ S
such that b E D and nb= a. This completes the proof of Lemma 1.1.
Consider tile reduced group S, = S/Sd (which we identify with a sub-
,re'cup of S). For each prime p, define a "semi-norm'" I I t, on S, by:
lalp = 0 ifp n divides a tbr every n > 0; otherwise, lalp = p-n if n is the
largest integer _> 0 such that pn divides a. Define, for every a ~ S r,
lal = p~l lalp 2 -p" this defines a norm oil S r, i.e.,
(i) lal = 0,=~ a = 0,
(ii) la+bl~ lal+lbl,
(iii) lal = I-al:
(i) follows from Lemma 1.1 because ifa ~ S r and lat = 0 then p" divides
a for every prime p and every n 2 0; hence a ~ S r n S a = {0},
It is easy to see that a sequence (an )n<,~ ~ti S r is Cauchy with respect
to 1 I if and only if it is Cauchy with respect to every semi-norm i :o1 and